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Profinite Structures and Dynamics
"... this paper with a few minor adjustments. For substructures we take subsets such that whenever an operation is defined on elements of the subset then the resulting value is also in the subset. For a homomorphism, whenever an operation is defined on elements of the domain, the corresponding operation ..."
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this paper with a few minor adjustments. For substructures we take subsets such that whenever an operation is defined on elements of the subset then the resulting value is also in the subset. For a homomorphism, whenever an operation is defined on elements of the domain, the corresponding operation should also be defined on their images and the usual relation (1) should hold. We assume further that there are unary relations in the language which are interpreted in structures so as to form partitions of their universes (into sorts in the language of computer science) and so that all operations take their arguments in one sort and all their values are also of a single sort. Note that this is a nontrivial restriction. It allows us to define products of structures as subsets of the Cartesian product consisting of elements in which all components have the same sort, and then define operations and relations componentwise. Profinite structures are defined as in the case of fullydefined operations and free profinite structures may be constructed by taking projective limits, which in turn are realized as appropriate substructures of products of finite structures
DECIDABILITY AND TAMENESS IN THE THEORY OF FINITE SEMIGROUPS
"... Abstract. This is a survey of results of the profinite approach to decidability problems in the theory of finite semigroups. ..."
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Abstract. This is a survey of results of the profinite approach to decidability problems in the theory of finite semigroups.